In mathematics, Milnor K-theory[1] is an algebraic invariant (denoted
) defined by John Milnor (1970) as an attempt to study higher algebraic K-theory in the special case of fields.
It was hoped this would help illuminate the structure for algebraic K-theory and give some insight about its relationships with other parts of mathematics, such as Galois cohomology and the Grothendieck–Witt ring of quadratic forms.
Before Milnor K-theory was defined, there existed ad-hoc definitions for
of a commutative ring, it was expected there should be an infinite set of invariants
This led to much study and as a first guess for what this theory would look like, Milnor gave a definition for fields.
His definition is based upon two calculations of what higher K-theory "should" look like in degrees
in general has a more complex structure, but for fields the Milnor K-theory groups are contained in the general algebraic K-theory groups after tensoring with
Note for fields the Grothendieck group can be readily computed as
since the only finitely generated modules are finite-dimensional vector spaces.
Also, Milnor's definition of higher K-groups depends upon the canonical isomorphism (the group of units of
) and observing the calculation of K2 of a field by Hideya Matsumoto, which gave the simple presentation for a two-sided ideal generated by elements
Milnor took the hypothesis that these were the only relations, hence he gave the following "ad-hoc" definition of Milnor K-theory as The direct sum of these groups is isomorphic to a tensor algebra over the integers of the multiplicative group
modded out by the two-sided ideal generated by: so showing his definition is a direct extension of the Steinberg relations.
is nilpotent, which is a powerful statement about the structure of Milnor K-groups.
to Bloch's Higher chow groups which induces a morphism of graded rings
With these, and the fact the higher Chow groups have a ring structure
giving the relation to Quillen's higher algebraic K-theory.
Every element of Milnor K-theory can be written as a finite sum of symbols.
representing a generalization of Milnor K-theory with coefficients in an abelian group
Note that sections of this pre-sheaf are equivalent classes of cycles on
[6] The others are lifts from the classical Steenrod operations to motivic cohomology.
The full proof is in the appendix of Milnor's original paper.
Milnor K-theory plays a fundamental role in higher class field theory, replacing
Milnor K-theory fits into the broader context of motivic cohomology, via the isomorphism of the Milnor K-theory of a field with a certain motivic cohomology group.
[8] In this sense, the apparently ad hoc definition of Milnor K-theory becomes a theorem: certain motivic cohomology groups of a field can be explicitly computed by generators and relations.
A much deeper result, the Bloch-Kato conjecture (also called the norm residue isomorphism theorem), relates Milnor K-theory to Galois cohomology or étale cohomology: for any positive integer r invertible in the field F. This conjecture was proved by Vladimir Voevodsky, with contributions by Markus Rost and others.
Finally, there is a relation between Milnor K-theory and quadratic forms.
For a field F of characteristic not 2, define the fundamental ideal I in the Witt ring of quadratic forms over F to be the kernel of the homomorphism
[10] Dmitri Orlov, Alexander Vishik, and Voevodsky proved another statement called the Milnor conjecture, namely that this homomorphism